Theoretical investigation of the phase stability and elastic properties of TiZrHfNb-based high entropy alloys

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Theoretical investigation of the phase stability and elastic properties of TiZrHfNb-based high entropy alloys

J.H. Daiâ,b,⁎^{, W. Li}â^{, Y. Song}^b^{, L. Vitos}â,c,d

aApplied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, Stockholm SE-100 44, Sweden

bSchool of Materials Science and Engineering, Harbin Institute of Technology at Weihai, 2 West Wenhua Road, Weihai 264209, China

cDepartment of Physics and Astronomy, Division of Materials Theory, Uppsala University, Box 516, SE-75120 Uppsala, Sweden

dInstitute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, H-1525 Budapest, Hungary

H I G H L I G H T S

• The phase stability of equimolar TiZrHfNbX is closely connected to the d-occupation.

• The alloying elements have similar effects on the phase stability and elastic moduli of TiZrHfNb-based HEAs.

• The charge density at the Wigner-Seitz cell boundary has strong connections with the bulk moduli of TiZrHfNb- based HEAs.

G R A P H I C A L A B S T R A C T

Phase stability and elastic properties of TiZrHfNbX are studied by ﬁrst principles calculations.

a b s t r a c t a r t i c l e i n f o

Article history:

Received 13 May 2019

Received in revised form 28 June 2019 Accepted 10 July 2019

Available online 10 July 2019

First principles calculations are performed to study the effects of alloying elements (X = Al, Si, Sc, V, Cr, Mn, Cu, Zn, Y, Mo, Ta, W and Re) on the phase stability and elastic properties of TiZrHfNb refractory high entropy alloys.

Both equimolar and non-equimolar alloys are considered. It is shown that the calculated lattice parameters, phase stability and elastic moduli of equimolar TiZrHfNbX are consistent with the available experimental and theoretical results. The substitutions of alloying elements at Ti, Zr, and Hf sites with various contents show similar effects on the phase stability and elastic properties of the TiZrHfNb-based alloys. The substitutions on Nb site are found to generally decrease the stability of body centered cubic phase. Close connections between the charge densities at the Wigner-Seitz cell boundary and the bulk moduli of TiZrHfNb-based alloys are found. The present results provide a quantitative model for exploring the phase stability and elastic properties of TiZrHfNb-based alloys from the electronic structure viewpoint.

creativecommons.org/licenses/by/4.0/).

Keywords:

First principles High entropy alloys Phase stability Elastic properties

Data availability:

The raw/processed data required to reproduce theseﬁndings will be made available on request.

⁎ Corresponding author at: Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, Stockholm SE-100 44, Sweden.

E-mail address:daijh@kth.se(J.H. Dai).

https://doi.org/10.1016/j.matdes.2019.108033

Contents lists available atScienceDirect

Materials and Design

j o u r n a l h o m e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / m a t d e s

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1. Introduction

Due to the multi-principal element character and the slow diffusion kinetics, high entropy alloys (HEAs) often occur in single/dual phase with simple crystal structures. Compared to conventional alloys, HEAs often show excellent mechanical properties [1]. Composed of refractory elements, the so called refractory high entropy alloys (RHEAs) are promising for high-temperature applications, since many of them main- tain high strength up to 1600 °C [2,3].

The HfNbTaTiZr [4], HfNbTiZr [5], and HfMoNbTiZr [6] alloys are the well-known RHEAs [7] and own high yield strength at elevated temperature. The composing elements of RHEAs are often from groups IV–VI, which all show body centered cubic (bcc) structure at high temperature. Most RHEAs also occur in single bcc phase, and therefore they have low ductility at room temperature. Aluminum and silicon are often added in HEAs to adjust their ductility, oxidation properties and also the phase stability. Aluminum is a bcc stabilizing element for metallic Ti, Zr, and Hf and owns high solubility in them, therefore, the TiZrHfNbAl exhibits single bcc phase [8]. However, intermetallics such as AlxZr5/M5Si3can easily be formed as a second phase due to the strong interactions between Al and Zr [9,10] or Si and metallic elements [11–13]. The intermetallic compounds harden but often also embrittle the HEAs and make the processing more difﬁcult compared to that of the solid solution phase, which usually shows high ductility. Although design of appropriate intermetallic phases in HEAs is very challenging, it is a very effective way to strengthen HEAs. Recently, Tsai et al.

reviewed 142 intermetallic compounds containing HEAs, and found ﬁve most prevalent intermetallic phases [14].

A series ofﬁrst principles calculations have been carried out to study the phase stability and elastic properties of RHEAs [15–20]. The mechanical stability and intrinsic ductility can be estimated by evaluating the elastic constants. The special quasi-random structure (SQS) [21,22]

and coherent potential approximation (CPA) [23–25] are usually employed to model the chemical disordered alloys. By comparing the calculated results between SQS and CPA on HEAs, the local lattice distortions were found to have small effect on the lattice parameters and elastic constants [19,26] and thus for these properties CPA turned out to be an efﬁcient approach to describe the HEAs from ﬁrst principles.

Various parameters, such as the atomic size difference [27,28], valence electron concentrations (VEC) [7,29,30], entropy of mixing and enthalpy of mixing between alloying elements [27], are used to predict the phase stability of HEAs. The VEC is found to have connections with the ductility of HEAs. When it isb4.5, the RHEAs consisting of IV, V, and VI metals are intrinsically ductile [7]. The VEC has effects also on the phase stability, the face centered cubic (fcc) phases is stable when VEC is larger than 8, whereas the bcc phase is stable when VEC is smaller than 6.87 [30]. For the hexagonal close packed (hcp) phases, their VEC values are often in the range of (6.87, 8) [31,32]. Furthermore, the formation of solid solution in HEAs is very complex, and the common Hume-Rothery rules often fail. One example is that the increasing Al content stabilizes the bcc phase of CoCrCuFeNi [33] although the metallic Al occurs in the fcc structure.

To better understand the effect of composition on the elastic properties and phase stability of TiZrHfNb-based alloys, here we carry outﬁrst principles calculations for the equiatomic TiZrHfNbX HEA as well as for the non-equiatomic Ti_1−xZrHfNbXx, TiZr_1−xHfNbXx, TiZrHf_1−xNbXx, and TiZrHfNb_1−xX_xHEAs with X = Al, Si, Sc, V, Cr, Mn, Cu, Zn, Y, Mo, Ta, and W, and x varying between 0 and 1. We show that the d- occupation correlates well with the phase stability, and the charge densities at the Wigner-Seitz cell boundary have strong connections to the elastic moduli of the present systems.

The rest of the paper is organized as follows. InSection 2we present the computational methodology and numerical details.Section 3as- sesses the accuracy of our calculations, whereas results are presented and discussed inSection 4. The paper ends with conclusions.

2. Computational methodology

2.1. Total energy calculations

The exact mufﬁn-tin orbital (EMTO) [34,35] method is employed for theﬁrst principles calculations. All TiZrHfNbX systems were treated as disordered modelled by the coherent potential approximation [23,25,36]. The Perdew–Burke–Ernzerhof (PBE) [37] was chosen as the exchange-correlation functional. The screened impurity model parameter of CPA was set to 0.6 as used by Tian et al. [18]. The one- electron Kohn-Shan equations were solved within the scalar- relativistic approximation and soft-core scheme.

2.2. Elastic constants

The elastic constants of single-crystal can be obtained byﬁtting the strain-energy or strain-stress relations. For the cubic structure, there are only three independent elastic constants, C11, C12, and C44. The volume-conserving distortions are applied on the cubic crystal to calcu- late the tetragonal shear modulus C′ = (C11− C12) / 2, and C44. The orthorhombic distortion results in a strain tensor,

Do¼ δo

0 0

0−δo0 0 01

1−δ²o

0 BB

@

1 CC

A ð1Þ

The C44can be extracted by the monoclinic distortion,

Dm¼

1 δ^m 0

δ^m 1 0

0 0 1

1−δ²m

0 BB

@

1 CC

A ð2Þ

The bulk modulus can be derived byﬁtting the equation of states, in which the Morse-type function is used [38]. The poly-crystalline elastic moduli are obtained from the three cubic elastic constants using the Voigt and Reuss bounds [39–41]. According to Voigt approach, the bulk and shear moduli (BVand GV) are derived from:

BV¼C11þ 2C12

3 ð3Þ

GV¼ðC11−C12þ 3C44Þ

5 ð4Þ

The bulk modulus extracted from the Reuss approximation (BR) is equal to the BV, and the shear modulus GRcan be expressed as:

GR¼ 5 Cð 11−C12C44Þ

4C44þ 3 Cð 11−C12Þ ð5Þ

The Young's modulus E can be calculated using the bulk (B) and shear (G) moduli based on Hill approximation, which is the arithmetic average of the Voigt and Reuss bounds, viz.

E¼ 9BG

3Bþ G ð6Þ

2.3. Numerical details

The EMTO basis set included s, p, d, and f orbitals. The 4d, 5s states of Nb, the 3s and 3p states of Al and Si, the 3d and 4s states of Sc, Ti, V, Cr, Mn, Cu, and Zn, the 4d and 5s states of Y, Zr, and Mo, the 5d and 6s states of Ta, W, and Re, and the 5d, 6s, and 4f states of Hf were treated as valence states. The number of orbitals in the full charge density calculation of the total energy was chosen 8.

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To ensure the calculation accuracy, the k-meshes of cubic (bcc and fcc) and hcp phases were chosen as 21 × 21 × 21 and 31 × 31 × 19, respectively. For the orthorhombic and monoclinic distortions, the k- meshes were 31 × 31 × 31 and 27 × 27 × 37, respectively. For the equation of state, ten points were considered around the equilibrium Wigner-Seitz radius for the bcc and fcc phases. For the hcp phase, the maps of seven different c/a ratios and ten Wigner-Seitz radii were calculated toﬁnd the equilibrium lattice parameters.

To make sure that the valence states are correctly captured below the Fermi energy level, proper energy contour should be chosen. The energy contour should be large enough to include all the valence states, but should not include any high-lying core states. For the 5d transition metals of Hf, Ta, W and Re, their f states should be carefully treated in order to get accurate results. The 4 f states of atomic Hf and Ta have close energy (−1.4 to −1.8 Ry relative to the vacuum level), while the 4 f states of atomic W and Re have much lower energy (−2.4 to

−3.2 Ry). Therefore, the 4 f states of Hf may interact with that of Ta in

equimolar TiZrHfNbTa system, and affect the distributions of valence states. In order to obtain correct description of the valence states, the energy contour was chosen as 1.0 Ry for the systems without Ta. For the Ta containing systems, the energy contour was set to 0.94 Ry to se- cure the 4f states of Hf within the contour as shown in the total density of states of equimolar TiZrHfNbX systems (Fig. 1(a)). The bonding peaks of alloying elements can be distinguished from the local density of states as shown inFig. 1(b). The sharp bonding peaks around−0.8 Ry are the 4f states, which means the energy contour should be larger than this value in order to include the Hf 4f states in the valence band. The bonding peaks around−0.4 Ry are 3d states of Zn, while the board bonding area between−0.7 and −0.4 Ry is contributed by Si.

3. Assessing the accuracy

Generally, the numerical parameters used in this study are similar with those used for the equimolar TiZrNbV(Mo) systems [18]. In order

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

0 20 40 60 80

Total density of states (states/Ry)

Energy relative to the Fermi energy level (Ry) TiZrHfNb TiZrHfNbAl TiZrHfNbSi TiZrHfNbSc TiZrHfNbV TiZrHfNbCr TiZrHfNbMn TiZrHfNbCu TiZrHfNbZn TiZrHfNbY TiZrHfNbMo TiZrHfNbTa TiZrHfNbW TiZrHfNbRe Zn

Hf

Cu

Si

(a)

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

0 10 20 30 40

Cr Mn Cu Zn Y Mo Ta W Re

Partial density of states (states/Ry)

Energy relative to the Fermi energy level (Ry) Ti Zr Hf Nb Al Si Sc V

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Fig. 1. Density of states of bcc equimolar TiZrHfNbX alloys. Shown are the (a) total density of states and (b) partial density of states. The solid vertical line at 0.0 marks the Fermi level.

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to assess the calculation accuracy for the equimolar TiZrHfNbX systems, we collected the lattice parameters and elastic constants of these alloys from literatures and show them inTable 1. For the equimolar TiZrNbV (Mo) systems, the calculated lattice parameters are generally close to the values calculated by Tian et al. [18]. Due to the different exchange- correlation approximations, the calculated bulk modulus, C₁₁and C₁₂ values by the generalized gradient approximations (GGA) in this study are smaller than those obtained by the local density approximation (LDA) [19,42] for the equimolar TiZrNbV, TiZrHfNb, and TiZrHfNbV systems. However, the GGA and LDA results are close to each other for the equimolar TiZrHfNbTa. For equimolar TiZrHfNb, the bulk modulus calculated via PBE is much closer to the experimental measurements than that by LDA. For equimolar TiZrHfNbV, the experimental lattice parameters scatter, and the lattice parameter is 3.3476 Å estimated via Vegard's law [19]. Applying PBE to equimolar TiZrHfNbV we obtained a lattice parameter that is close to the above estimated value and also to the experimental value from Ref. [43]. However, there are some dis- crepancies between the calculated and experimental data, which may be caused by temperature effects or synthesizing process. Furthermore, the LDA and GGA methods used to describe the exchange-correlation interactions also bring different elastic constants for Hf containing alloys [46]. Considering above results and the high accuracy of GGA-PBE for the Hf based alloys [46], the PBE functional is used in our calculations.

4. Results and discussion

4.1. Equimolar TiZrHfNbX systems

To explore the effect of alloying element X on the phase stability of equimolar TiZrHfNb, inFig. 2we show the calculated equilibrium Wigner-Seitz radii, the energies of the fcc and hcp phases relative to that of the bcc phase, and the equilibrium bulk moduli.

According toFig. 2(b), the alloying elements considered here can be divided to three groups: Y, Sc, and Ta enlarge the Wigner-Seitz radius of the host, Re reduces the Wigner-Seitz radius, and Si, V, Cr, Mn, Cu, Zn, Mo, and W have weak inﬂuence on it. We observe that the variations of the equilibrium Wigner-Seitz radii of equimolar TiZrHfNbX alloys follow the same trend as the atomic radii of the alloying elements X. In other words, the atomic radius of the dopant X is the main factor that controls the volume of the equimolar TiZrHfNbX.

Due to the effects of alloying elements X on the equilibrium volumes of equimolar TiZrHfNbX systems, the bulk modulus of alloys will be changed correspondingly. Generally, the larger volume the alloy has,

the smaller bulk modulus is expected. Exceptions are the Ta containing alloys. The Mo, Ta, W, and Re containing alloys have larger bulk moduli than the host alloy, while in other cases the bulk moduli are rarely changed by alloying. The bulk moduli of metallic Mo, Ta, W, and Re are 2–3 times larger than that of metallic Ti, Zr, Hf and Nb. Therefore, it is expected that alloying with Mo, Ta, W, or Re increases the bulk moduli of equimolar TiZrHfNb system.

In order to compare the relative stability of the bcc, fcc, and hcp phases of the equimolar TiZrHfNbX systems, the energy difference of the fcc (hcp) phase relative to the bcc phase are plotted inFig. 2(c).

The total energy refers to the total energy as deﬁned in Density Func- tional Theory (sum of kinetic, Hartree, external and exchange- correlation terms). The internal energy is the total energy in this work.

Results located within shaded area means the corresponding element is hcp stabilizing element for the equimolar TiZrHfNb alloys. Except the Sc and Y containing systems, all other alloys show that the bcc phase is energetically stable, especially of the Mo, Ta, W and Re bearing alloys. Accordingly, Mo, Ta, W, and Re are strong bcc stabilizing elements for the TiZrHfNb-based alloys. On the other hand, Sc and Y are hcp stabilizing elements, which is likely to be due to the fact that the two elements have hcp structure in their ground state. The bcc structure of equimolar TiZrHfNbRe is more stable than the fcc and hcp structures according our calculations, although the metallic Re has the hcp structure. Recently, the equimolar TiZrHfNbRe was experimentally veriﬁed to have a single bcc phase by Marik et al. [47]. As shown in supplementary information, the precipitation phases can be easily formed in Al, Si, Cr, Cu, and Re alloyed TiZrHfNb alloys. We have therefore classiﬁed these alloying elements into two types. The precipitation elements (Al, Si, Cr, Cu, and Re), and solid-solution elements (Sc, V, Mn, Zn, Y, Mo, Ta, and W).

The d-occupation (n_d) was found to play important role on crystal structure of transition metals by Skriver [48]. Increasing the d- occupation, the transition metals follow the hcp→ bcc → hcp → fcc structural sequence. The hcp stability range for transition metal is 1

≤ nd≤ 2 and 6 ≤ nd≤ 7, and the transition metal exhibits fcc structure for 8.5≤ nd≤ 10. Transition metals have energetically stable bcc structure for all other ndvalues. Recently, Al-Zoubi et al. reported that the 4d transition binary alloys obey this theory [49]. The d-occupation of equimolar TiZrHfNb is 2.4 by using number of d electrons of element given by Skriver [48]. Since Sc and Y have one d electron, the d- occupation of equimolar TiZrHfNbSc(Y) decreases to 2.1 approaching the hcp stability range. Indeed, the energetically stable structure of Table 1

The calculated and experimental lattice parameters (Å), elastic constants and bulk moduli (GPa) of selected equimolar RHEAs.

System a C11 C12 C44 B

TiZrNbV Present 3.287 167.5 89.3 60.7 115.34

Ref. [18] 3.290 165.8 92.8 50.5 117.1

Ref. [42] 3.320 192.8 127.9 127.9 149.4

Ref. [17] 3.303 159.8 114.3 18.5 129.5

TiZrNbMo Present 3.303 211.6 95.7 57.1 134.3

Ref. [18] 3.306 209.6 98.8 49.9 135.8

TiZrHfNb Present 3.419 142.1 95.5 76.9 111.0

Ref. [42] 3.509 154.3 127.9 56.7 136.7

Ref. [20] – – – – 112.9

TiZrHfNbV Present 3.349 145.3 85.4 70.0 105.4

Ref. [19] 3.401 149.5 115.1 55 126.6

Ref. [43]^a 3.358 – – – –

Ref. [19]^a 3.385 – – – –

TiZrHfNbV Ref. [40] 3.428 170.2 130.8 51.3 143.9

TiZrHfNbTa Present 3.388 188.0 118.1 82.4 141.4

Ref. [16] 3.329 229.7 132.4 20.7 140.1

– 3.41 [45]â 178 [44]â 112.9 [44]â 28 [44]â 134.6 [44]â

Ref. [19] 3.457 160.2 124.4 62.4 136.3

Ref. [20] – – – – 125.4

aExperimental value.

Fig. 2. The calculated (a) bulk moduli, (b) equilibrium Wigner-Seitz radii, and (c) total energies of the fcc (hcp) phase relative to the bcc phase for the equimolar TiZrHfNbX alloys shown in terms of alloying element X. The dashed lines denote the parameters calculated for the equimolar TiZrHfNb alloy. The shaded area marks the hcp stabilizing zone.

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equimolar TiZrHfNbSc(Y) is hcp as shown inFig. 2. Although Cu and Zn have more d electrons, the average ndof equimolar TiZrHfNbCu(Zn) is 3.5≤ nd≤ 5 which is still within the bcc zone. The ndof other equimolar TiZrHfNbX systems is far within the bcc regime, and these systems indeed exhibit stable bcc structure. We conclude that the phase stability of equimolar TiZrHfNbX systems follows with good approximation the classical d-occupation theory.

The elastic constants of bcc equimolar TiZrHfNbX are shown inFig. 3.

The mechanical stability can be estimated by the Born-Huang criteria, e.g., C11, C12, C44N 0, and C′ = (C11− C12) / 2N 0. All studied equimolar TiZrHfNbX systems meet these stability criteria. Therefore, except the energetically unstable bcc equimolar TiZrHfNbSc(Y) systems, other studied equimolar TiZrHfNbX have mechanically and thermodynami- cally stable bcc structure. Most equimolar TiZrHfNbX systems show similar elastic moduli with exceptions of the Mo, Ta, W and Re containing alloys. They have much larger C₁₁compared to other systems, and therefore, their Young's moduli are also large.

4.2. Non-equimolar TiZrHfNbX systems

As shown in above discussions, our CPA calculations about equimolar TiZrHfNbX systems are consistent with the available theoretical and experimental results. In order to investigate the inﬂuence of substitution of a particular component by the alloying elements on the phase stability and elastic properties of TiZrHfNb alloy, in the following we consider the non-equimolar TiZrHfNbX systems. The X alloying element individually substitutes for Ti, Zr, Hf, and Nb of TiZrHfNb alloy, denoted as Ti_1−xZrHfNbX_x, TiZr_1−xHfNbX_x, TiZrHf_1−xNbX_x, and TiZrHfNb_1−xX_x, respectively. For simplicity, in the following we refer to the above substitutional alloys as“non-equimolar TiZrHfNbX” alloys. The results are shown inFigs. 4 to 6. The content of X varies in the range of 2 to 22.5 at.% with increment of 2 at.%. When x is 1, the system becomes a new quaternary alloy.

As expected, the Ti1−xZrHfNbXx, TiZr1−xHfNbXx, and TiZrHf1−xNbXx

systems show similar phase stabilities, due to the similar chemical properties of Ti, Zr, and Hf elements. The relative stability of bcc, fcc, and hcp phases are not greatly changed by the substitutions of X. Similar with the equimolar TiZrHfNbX systems (Fig. 2), the Sc and Y elements again decrease the bcc stability relative to the fcc and hcp structures.

The Ti_1−xZrHfNbXx, TiZr_1−xHfNbXx, and TiZrHf_1−xNbXx (X = Sc, Y) systems stabilize in the hcp structure, when the contents of Sc or Y are larger than 18 at.% and 22 at.%, respectively.

With increasing content of X (X = Mo, Ta, W and Re), the bcc structure of Ti_1−xZrHfNbXx, TiZr_1−xHfNbXx, and TiZrHf_1−xNbXxsystems becomes more and more stable relative to the fcc and hcp structures. The stability of fcc Ti_1−xZrHfNbXx, TiZr_1−xHfNbXx, and TiZrHf_1−xNbXxsys- tems is strengthened with increasing content of X (X = Al, Si, Sc, Cu, Zn and Y), but their bcc structure is still the most stable one. The Nb element is strong bcc stabilizing element, and therefore, substitution of Nb by alloying element X will lower the stability of TiZrHfNb_1−xXx

with bcc structure comparing to the host TiZrHfNb. Exceptions are Mo, W, and Re systems, which strengthen the stability of the bcc phase since they have larger d-occupation comparing to Nb. The hcp structure is more stable than the bcc structure when the contents of Al, Sc, Mn, Cu, Zn, and Y are high, especially for the Sc and Y containing systems. When their contents are larger than 6 at.% and 8 at.%, respectively, the hcp structure of TiZrHfNb_1−xXx(X = Sc, Y) will be more stable than their bcc and fcc structures.

The phase stability is found to be closely connected to the d- occupation for non-equimolar TiZrHfNbX alloys as shown inFigs. 2 and 5. When Al, Si, Sc, and Y substitute for Nb, the d-occupation decreases to 1.5 within the hcp stability region. The d-occupations of other non-equimolar TiZrHfNbX alloys are in the region of (2.4, 4.375), which is within the bcc stabilizing zone. Our total energy calculations fully support this conclusion as shown inFig. 4.

In order to study the structure dependence of d-occupation number and the inﬂuence of the substitutional alloying element X, the self- consistent d-occupation number (snd) of fcc and hcp structures relative to bcc structure of non-equimolar TiZrHfNbX alloys are plotted inFig. 5.

The self-consistent d-occupation numbers for every structure (bcc, fcc, and hcp structure) are extracted from EMTO calculations. We recall that the actual d-occupation (snd) and the d-occupation based on atomic electron numbers (nd) may differ due to the complex interactions between alloy components. The sn_dof bcc equimolar TiZrHfNb alloy is 2.87 (compared to 2.4 obtained for nd). The self-consistent s, p, and f electrons are almost unchanged compared to a wide variety of d electrons in studied systems. As shown in the bottom panel ofFig. 5, the sndof Ti_1−xZrHfNbXx, TiZr_1−xHfNbXx, and TiZrHf_1−xNbXxsystems are almost the same. It is a little bigger than that of corresponding TiZrHfNb1−xXxsystems, since the Ti, Zr, and Hf have fewer d electrons than that of Nb. The top two panels show the difference of snd(fcc/

hcp) relative to snd(bcc) of non-equimolar TiZrHfNbX alloy, denoting asΔsnd. AllΔsndare negative indicating the bcc structure has more d electrons than that of hcp or fcc structure. Therefore, the bcc structure of non-equimolar TiZrHfNbX alloy has more d electrons than the corresponding hcp and fcc structures. The actual d-occupation is a good indi- cator for the phase stability of the present RHEAs.

Fig. 6shows the elastic moduli of the non-equimolar TiZrHfNbX alloys. Weﬁnd that GH, EHand C′ change almost linearly with the amount of X. The elastic moduli of V, Cr, Mn, Cu, Mo, Ta, W, and Re containing systems are gradually increased with increasing X content, especially in the Mo, Ta, W, and Re containing systems. Due to the similar chemical properties of Ti, Zr, and Hf elements, the Ti1−xZrHfNbXx, TiZr1−xHfNbXx

and TiZrHf_1−xNbXxsystems show close elastic moduli with the exceptions of TiZr_1−xHfNbCr(Mn)x. The elastic moduli of TiZrHfNb_1−xXxsys- tems (except X = Mo, Ta, W, and Re systems) decrease with increasing X content, especially when X = Al, Si, Sc, Zn, and Y. The elastic moduli decrease sharply with increasing X content, and the C′ becomes negative at high contents of Sc(Y) containing systems indicating that these systems are mechanically unstable.

The ratio of bulk to shear moduli (B/G) is a good parameter to describe the intrinsic ductility of metallic alloys.Fig. 7shows the B/G values of non-equimolar TiZrHfNbX systems. Generally, the concentration of X shows similar effect on B/G ratio for Ti_1−xZrHfNbXx, TiZr1

−xHfNbXx, TiZrHf_1−xNbXx, and TiZrHfNb_1−xXxsystems. The B/G value of TiZr1−xHfNbTaxsystems increases with increasing Ta content. The Al, Si, Sc, Zn, and Y element can obviously improve the intrinsic ductility of TiZrHfNb_1−xXxsystems, and V, Cr, Mn, Mo, W and Re decrease the Al Si Sc V Cr Mn Cu Zn Y Mo Ta W Re --

0 50 100 150 200

250 C₁₁

C₁₂ C₄₄

Elastic constants and moduli (GPa)

X in equimolar TiZrHfNbX G_H

E_H C'

C' G_H C₄₄ C₁₂ C₁₁

Fig. 3. Elastic constants C11, C12, C44, Young's (E) and shear moduli (G, C′) (GPa) of bcc equimolar TiZrHfNbX alloys. The dashed lines denote the elastic constants and moduli of the equimolar TiZrHfNb alloys.

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intrinsic ductility of all studied systems, especially Cr and Mn. Therefore, the Al, Si, Sc, Zn and Y elements may be helpful for improving the intrinsic ductility of TiZrHfNb based alloys, while the V, Cr, Mn, Mo, W and Re element should be avoided in term of ductilization. The inﬂuence of Al on the ductility of TiZrHfNb based alloys is complicated and depends on the compositions of HEAs. For the AlxHfNbTaTiZr (x = 0, 0.3, 0.5, 0.75, and 1.0) composed of single bcc phase, the Al addition reduces its ductility [50]. However the NbTiVTaAlx(x = 0, 0.25, 0.5 and 1.0) alloys with single bcc structure own high ductility (no fracture under 50%

strains) [51]. The Al5(HfNbTiZr)95exhibits single bcc structure and has elongation of 31.5%, which is excellent ductility for bcc HEAs [52]. Ac- cording to our calculations, we conclude the intrinsic ductility of AlTiZrHfNb is good, although the ductility is linked with many factors, such as preparation, processes and etc.

The valence electron concentration has connections with the elastic moduli of metals [26], but it may not be a good parameter for TiZrHfNb based HEAs. Metallic V, Nb, and Ta have similar valence states, but they have obviously different effects on the elastic properties as shown in Figs. 3 and 6. Miedema et al. proposed the concept of charge density at the Wigner-Seitz cell boundary,ρWS, and built a relationship with the bulk modulus [53],

ρWS¼ 0:82 10⁻⁴ðB=VmÞ¹⁼²; ð7Þ

where Vmdenotes the atomic volume in unit of cm³/(g atom), andρWS

is expressed in e/Å³. Later, Cheng et al. improved the relationship based Fig. 4. Total energy (eV) of the fcc (hcp) phase relative to the bcc phase of non-equimolar TiZrHfNbX systems. The x axis show the X element and its contents (one minor tick denotes 2 at.

%). Shown are results for (a) Ti1−xZrHfNbXx, (b) TiZr1−xHfNbXx, (c) TiZrHf1−xNbXx, and (d) TiZrHfNb1−xXxsystems.

Fig. 5. The number of calculated d electrons snd(e) for the non-equimolar TiZrHfNbX alloys. HereΔsnd(hcp-bcc) (Δsnd(fcc-bcc)) means the sndof hcp (fcc) of the non- equimolar TiZrHfNbX systems relative to that of bcc structure. The dashed line denotes the sndof equimolar TiZrHfNb alloys.

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onﬁrst principles calculations for fcc and bcc metals [54],

B¼0:895 10⁸

n a³ρ²WS; ð8Þ

where a is the lattice parameter in unit of Å, B is the bulk modulus in unit of kg/cm², and the n means the number of atoms in the primitive cell. For ternary alloys, Li et al. proposed a similar relationship [55],

B¼ ρ²WSV; ð9Þ

where the volume (V) of alloys is deﬁned as the molar volume of intermetallic compound.

The relationships between the bulk modulus of non-equimolar TiZrHfNbX alloys and the charge density at the Wigner-Seitz cell boundary are shown inFig. 8. The bulk modulus can be approximated as

Be∝0:0031 ρ²WS ð10Þ

in present study. It shows nearly perfect correlation between the calculated bulk modulus (Bc) through EMTO and the estimated one (Be) using ρWS. Exceptions are the Cr and Mn containing alloys as shown by red dashed circle inFig. 8. This deviation may be due to magnetism, in particular to the high local magnetic moments of Cr (2.9μBper Cr atom) and Mn (3.4μBper Mn atom) in non-equimolar TiZrHfNbX (X = Cr 0

20 40 60 80 100 120 140 160 180 200

Al Si Sc V Cr Mn Cu Zn Y Mo Ta W Re G_H

E_H C'

Elastic moduli (GPa)

Ti_1-xZrHfNbX_x (a)

0 20 40 60 80 100 120 140 160 180 200

E_H C'

TiZr_1-xHfNbX_x (b)

0 20 40 60 80 100 120 140 160 180 200

E_H C'

TiZrHf_1-xNbX_x (c)

-20 0 20 40 60 80 100 120 140 160 180

E_H C'

TiZrHfNb_1-xX_x

Sc Y

(d)

Fig. 6. The Young's (E) and shear moduli (G, C′) (GPa) of non-equimolar TiZrHfNbX alloys in bcc phase, and the x axis shows the X element and its content (one minor tick denotes 2 at.%).

(a) Ti1−xZrHfNbXx, (b) TiZr1−xHfNbXx, (c) TiZrHf1−xNbXx, and (d) TiZrHfNb1−xXx.

1 2 3 4 5 6

Al Si Sc V Cr Mn Cu Zn Y Mo Ta W Re Ti_1-xZrHfNbX_x

TiZr_1-xHfNbX_x TiZrHf_1-xNbX_x TiZrHfNb_1-xX_x

B/G

TiZrHfNbX based alloys

Fig. 7. The B/G of the bcc non-equimolar TiZrHfNbX alloys. The x axis shows the X element and its content (one minor tick denotes 2 at.%).

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and Mn) alloys. Their charge density distributions are therefore affected by magnetic splitting, and thus the total densityρWSis not a good parameter any longer to describe the bulk modulus. When the content of alloying element is 25 at.% (x = 1), there is considerable dis- crepancy between the calculated and estimated bulk modulus as shown by magenta dashed circle inFig. 8. The TiZr1−xHfNbTax (X = Ta) systems show larger value of Bc comparing to that of Be. For the other systems, the calculated bulk moduli are in line with the estimated ones.

5. Conclusions

We have studied the phase stability and elastic properties of TiZrHfNb based refractory HEAs based onﬁrst-principle alloy theory.

The phase stability of equimolar TiZrHfNbX is closely connected to the d-occupation. The equimolar TiZrHfNbSc(Y) exhibits hcp structure, and other studied equimolar TiZrHfNbX systems are energetically and mechanically stable in the bcc structure. The alloying elements are found to have similar effects on the phase stability and elastic moduli of Ti_1−xZrHfNbXx, TiZr_1−xHfNbXx, and TiZrHf_1−xNbXxnon-equimolar systems. For the TiZrHfNb1−xXxsystems, the stability of the bcc structure is generally decreased with increasing X content, except the Mo, W, and Re containing systems which prefer a stable bcc structure. The bcc structure is the most stable one, and it owns the smallest bulk modulus in most TiZrHfNbX systems, while the hcp phase has the largest value. The present work conﬁrms that the charge density at the Wigner-Seitz cell boundary has strong connections with the bulk moduli of TiZrHfNb-based HEAs.

CRediT authorship contribution statement

J.H. Dai: Investigation, Visualization, Data curation, Formal analysis, Writing original draft. W. Li: Investigation, Validation, Methodology. Y.

Song: Writing - review & editing. L. Vitos: Supervision, Project adminis- tration, Writing - review & editing.

Acknowledgments

The authors acknowledge the Swedish Research Council, the Swed- ish Foundation for Strategic Research, the Carl Trygger Foundation, the Swedish Foundation for International Cooperation in Research and Higher Education, the Hungarian Scientiﬁc Research Fund (OTKA 128229), National Basic Research Program of China (Grant No.

2016YFB0701301), and the China Scholarship Council forﬁnancial sup- ports. The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at Linköping.

Appendix A. Supplementary data

Supplementary data to this article can be found online athttps://doi.

org/10.1016/j.matdes.2019.108033.

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